Abstract
Let \({\mathfrak {g}}\) be a finite-dimensional perfect Lie algebra over a field k of characteristic 0. In infinite-dimensional Lie theory we encounter Lie algebras of the form \({\mathfrak {g}}\otimes _k R\), where R is a k-ring (usually a Laurent polynomial ring in finitely many variables over k), and étale twisted forms \({\mathcal {L}}\) of \({\mathfrak {g}}\otimes _k R\). Thus \({\mathcal {L}}\) is an R-Lie algebra that becomes isomorphic to the S-Lie algebra \({\mathfrak {g}}\otimes _k S\) after some étale cover base ring extension S/R. The interesting infinite-dimensional Lie algebras are “built” out of \({\mathcal {L}}\) by adding a centre \({\mathcal {Z}}\) and a Lie algebra of derivations \({\mathcal {D}}\) (the affine Kac-Moody Lie algebras are the simplest examples). \({\mathcal {D}}\), which determines \({\mathcal {Z}}\), is a Lie subalgebra of \({\text {Der}}_{k}(\mathcal {L})\) of \({\mathcal {L}}\). The understanding of this last Lie algebra is crucial. While the R-Lie algebra \({\mathcal {L}}\) can be given by étale descent, the same cannot be openly said about \({\text {Der}}_{k}({\mathcal {L}})\) since it is not an R-Lie algebra. In the present paper we give such a descent presentation within the more general framework of relative R/k-sheaves of Lie algebras that we believe is of independent interest.
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Notes
An extension \(0 \rightarrow {\mathcal {C}} \rightarrow {\tilde{{\mathcal {L}}}} \rightarrow {\mathcal {L}}\rightarrow 0\) of \({\mathcal {L}}\) is called central, if \({\mathcal {C}}\) is contained in the center of \({\tilde{{\mathcal {L}}}}\). Such an extension is universal, if it maps uniquely to any other central extension (of the same Lie algebra \({\mathcal {L}}\)).
For convenience we will ignore set theoretical considerations that are inherent to our work with categories. The reader interested on this can make the necessary modifications to what we have written following [5]. For example R should be a model, and so should the rings of \(R_{{\text {alg}}}\).
In this category all ring homomorphisms are étale.
Our R is k in [5]. Their base k-sheaf \({\mathfrak {S}}\) will for us always be R, or more precisely \({\mathfrak {S}}{\mathfrak {p}}(R)\). We will later denote by k a subring of R to create the “relative” Lie algebra context that is central to our work. A typical example of this is the R-module of Kähler differentials \(\Omega _{R/k}\) as explained in Example 3.3 below.
This is a right action of the group \({\mathfrak {G}}(S)\) on the set \({\mathfrak {X}}(S)\) which is functorial on S in \(R_{\acute{{\text {e}}}{\text {t}}}\).
We again leave the set theoretic considerations to the interested reader.
These are pointed sets, and “kernel” refers to the subset of elements map** to the special point in the target; in this context the special points in question are the equivalence classes of trivial torsors.
It is the algebraic geometric analogue of a fibre bundle associated to a given principal bundle in the theory of topological groups.
Our \(H^1\) and \({\mathfrak {X}}\wedge ^{\mathfrak {G}}{\mathfrak {H}}\) notation is now standard. In [5] they are denoted by \({\tilde{H}}^1\) and \({\mathfrak {X}}\vee ^{\mathfrak {G}}{\mathfrak {H}}\) respectively.
Prime examples are functors of the form \({\mathfrak {D}}(M):S \mapsto M \otimes _R S\) where M is some R-module. \({\mathfrak {D}}_R\)-modules of this form are often referred to as quasi-coherent and they are automatically sheaves.
The S-module structure on \({\mathfrak {L}}(T)\) is naturally the restriction induced by \(f:S \rightarrow T\).
These are called \(R-L\)-modules in [9].
There is some room for ambiguity in this notation in cases where L is both an associative algebra and a Lie algebra by means of the commutator map of the associative structure; we trust that this will not lead to confusion.
Heretofore we denote by \(\chi _s\) the scalar multiplication by s.
See Sect. 4 below.
See Remark 3.10 below.
We call a subalgebra of an R/k-Lie algebra characteristic (resp. fully characteristic) if it is preserved by all automorphisms (resp. endomorphisms) of the R/k-Lie algebra.
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We would like to thank the referee for the many useful suggestions on how to improve our manuscript.
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H. Guo: Partially supported by NSFC (No. 11901224) and the Fundamental Research Funds for the General Universities (No. 30106220276).
J. Kuttler: Partially supported by an NSERC Discovery Grant.
A. Pianzola: The author is grateful to NSERC and Conicet for their continuous support.
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Guo, H., Kuttler, J. & Pianzola, A. Descent study of the Lie algebra of derivations of certain infinite-dimensional Lie algebras. manuscripta math. 173, 1195–1215 (2024). https://doi.org/10.1007/s00229-023-01483-6
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DOI: https://doi.org/10.1007/s00229-023-01483-6